Flight instructor reported an NMAC event during departure for a training flight with an intruder aircraft entering the traffic pattern incorrectly. Flight instructor altered the landing pattern to avoid a conflict.

Narrative

After making a radio call that I was departing ZZZ towards the north; I took off from ZZZ northbound in the PA-18-150 on straight floats. I had one student on board with me training for his seaplane rating and no other souls on board. As we began our climb out on departure; I noticed the PT-17 over the airport on a western heading and heard his radio call that he intended to enter the pattern. He appeared to be about 1000 ft.; which would put that aircraft 500 ft. below the FAA's recommended midfield altitude when entering the pattern. The pilot of the PT-17 began turning left on a downwind earlier than I expected. It also became clear; with this action; that the pilot of the PT-17 was using the alternate downwind entry from midfield as opposed to the FAA's advised entry of a teardrop. The course the PT-17 was turning towards appeared to put them on course to pass over the top of me as I departed northbound. I assumed they didn't see me; so I started a turn roughly 30 degrees to the right and made a radio call that I had the PT-17 in sight and we were no longer a factor for each other. By the time we passed each other; my PA-18-150 was 500 ft. below the PT-17 and 500-1000 ft. away laterally. The pilot of the PT-17 broadcasted on the radio; 'I would call that a factor.' I told him my name and invited him to discuss it further if he wanted and he invited me to his office.Myself and my student had the aircraft in sight the entire time and I took action to not only remain well below them but also move laterally so we didn't pass directly underneath them. The PA-18-150 has excellent forward visibility and a window along the top of the cockpit making my upward visibility fantastic as well. In my and my student's opinion; we were far from interfering with the PT-17 or causing any concern of being too close. However; it has become clear that the PT-17 pilot mistakenly believes we passed very close to and directly underneath them; though he also claims his visibility is greatly impaired by the PT-17 airframe. One thing that could have assuaged the PT-17 pilot's concern would be for them to cross midfield at the FAA's recommended altitude; which would have put them 500 ft. higher than they were. That would have given them a full 1;000 ft. clearance from the departing PA-18-150. On my end; perhaps I could have leveled off or even descended; though I have concerns about buzzing the houses/road traffic off my departure end and remaining low over a congested area with few to no emergency options. The PT-17's radio call also sounded angry/sarcastic. Instead of pursuing further conversation with him to try and solve the negative interaction; I should have continued without further engagement. Though it was my intention to deescalate; it had the opposite affect and I wish I hadn't pursued further conversation.